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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Direct product ： ウィキペディア英語版 Direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. Examples are the product of sets (see Cartesian product), groups (described below), the product of rings and of other algebraic structures. The product of topological spaces is another instance. There is also the direct sum – in some areas this is used interchangeably, in others it is a different concept. == Examples == * If we think of $\backslash mathbb$ as the set of real numbers, then the direct product $\backslash mathbb\backslash times\; \backslash mathbb$ is precisely just the cartesian product, $\backslash$. * If we think of $\backslash mathbb$ as the group of real numbers under addition, then the direct product $\backslash mathbb\backslash times\; \backslash mathbb$ still consists of $\backslash$. The difference between this and the preceding example is that $\backslash mathbb\backslash times\; \backslash mathbb$ is now a group. We have to also say how to add their elements. This is done by letting $(a,b)\; +\; (c,d)\; =\; (a+c,\; b+d)$. * If we think of $\backslash mathbb$ as the ring of real numbers, then the direct product $\backslash mathbb\backslash times\; \backslash mathbb$ again consists of $\backslash$. To make this a ring, we say how their elements are added, $(a,b)\; +\; (c,d)\; =\; (a+c,\; b+d)$, and how they are multiplied $(a,b)\; (c,d)\; =\; (ac,\; bd)$. * However, if we think of $\backslash mathbb$ as the field of real numbers, then the direct product $\backslash mathbb\backslash times\; \backslash mathbb$ does not exist – naively defining $\backslash$ in a similar manner to the above examples would not result in a field since the element $(1,0)$ does not have a multiplicative inverse. In a similar manner, we can talk about the product of more than two objects, e.g. $\backslash mathbb\; \backslash times\; \backslash mathbb\; \backslash times\; \backslash mathbb\; \backslash times\; \backslash mathbb$. We can even talk about product of infinitely many objects, e.g. $\backslash mathbb\; \backslash times\; \backslash mathbb\; \backslash times\; \backslash mathbb\; \backslash times\; \backslash dotsb$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Direct product」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.